#7902826

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#50652	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,6)(7,8)(9,10)(11,20)(12,23)(13,24)(14,19)(15,16)(17,22)(18,21)(25,26)(27,34)(28,33)(29,30)(31,35)(32,38)(36,40)(37,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,21)(13,22)(15,19)(16,20)(17,18)(23,24)(25,27)(26,32)(28,39)(29,37)(30,36)(31,38)(33,40)(34,35);(1,3,5,4)(2,7,6,8)(10,16)(11,19)(12,22)(18,24)(25,28,31,30)(26,33,35,29)(27,36,38,39)(32,37,34,40);(1,3,5,4)(2,7,6,8)(9,12,15,22)(10,17,16,23)(11,21,19,13)(14,18,20,24)(25,29)(26,28)(27,37)(30,35)(31,33)(32,39)(34,36)(38,40);(1,3,5,4)(2,7,6,8)(9,13,15,21)(10,18,16,24)(11,22,19,12)(14,17,20,23)(25,30,31,28)(26,29,35,33)(27,36,38,39)(32,37,34,40);(1,4,5,3)(2,8,6,7)(9,14,15,20)(10,19,16,11)(12,18,22,24)(13,23,21,17)(25,27)(26,34)(28,36)(29,37)(30,39)(31,38)(32,35)(33,40)

Polycyclic presentation:

2^9|c^2=gh,d^2=ghi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7810022);